A transformer transfers electrical energy from one circuit to another through inductively coupled conductors, which are the coils. A change in current in a primary coil of a first circuit generates a time-dependent magnetic flux through a secondary coil in a second circuit. One of Maxwell's four equations provides:
            ∇              ×        E              =          -                        ∂          B                          ∂          t                      ,which can be rewritten in an integral form as:
            ∮              ∂                                  ⁢        S              ⁢          E      ·              ⅆ        l              =      -                            ∂                      Φ                          B              ,              S                                                ∂          t                    .      Thus, the time-dependent magnetic flux through the secondary coil induces an electromotive force (EMF) or in the secondary coil.
In many transformers, a ferromagnetic core is employed to guide the magnetic flux more effectively between the primary coil and the secondary coil. The ferromagnetic core is made of iron, cobalt, nickel, and/or other ferromagnetic materials, most of which have a density of at least 7.8 g/cm3. Thus, transformers with a ferromagnetic core tend to be heavy, in addition to being costly to manufacture.
An air core transformer is a transformer that does not have a ferromagnetic core. As the name suggests, the primary coil and the secondary coil are separated by air. While air core transformers tend to have a coupling coefficient k that deviates significantly from 1.0, air core transformers are significantly lighter and cheaper to manufacture than ferromagnetic core transformers having a comparable inductive coupling, as well as not as critically being dependent on alignment between the primary coil and the secondary coil. The coupling coefficient k refers to the ratio of the magnetic flux that cuts through both the secondary coil and the primary coil to the total magnetic flux that cuts through the primary coil. A transformer is “tightly coupled” if the coupling coefficient k is greater than 0.5, and is “loosely coupled” if the coupling constant k is equal to or less than 0.5.
Most air core transformers are loosely coupled transformers except for specially designed variants in which the secondary coil and the primary coil are designed to be located in close proximity with precise alignment. In loosely coupled air core transformers, the non-critical dependence of the performance of an air core transformed on the alignment between the primary coil and the secondary coil enables physical separation of an air core transformer into two movable parts. In other words, a first structural part including a primary coil can be physically displaced from a second structural part including a secondary coil, and subsequently put together without requiring a precise alignment therebetween for power transfer.
The potential to displace and reposition the secondary coil relative to the primary coil in a loosely coupled air core transformer can be exploited to inductively transfer power from a power outlet to an electrical vehicle. Methods of transferring power through inductive coupling are shown, for example, in U.S. Pat. No. 6,934,167 to Jang et al., U.S. Pat. No. 6,934,165 to Adler et al., and U.S. Pat. No. 6,418,038 to Takahama et al and in U.S. Patent Application Publication Nos. 2009/0322307 to Ide and 2009/0303753 to Fu et al. Prior art methods transfer power at a resonance frequency f0 of an air core transformer, which is given by:
            f      0        =          1              2        ⁢        π        ⁢                  LC                      ,in which L is the inductance of the circuit including the primary coil and C is the capacitance of the circuit including the primary coil. The resistance of the circuit including the primary coil is not considered in determining the resonance frequency f0, although the resistance of the circuit including the primary coil affects the Q-factor of the resonance. The circuit parameters of the secondary circuit including the secondary coil are selected to induce resonance at the resonance frequency f0, i.e., such that the product of the inductance and the capacitance of the secondary circuit matches the product of the inductance and the capacitance of the primary circuit.